Small-boundary reconstruction conjecture for discrete tomography

Let row sums be $\mathcal{R}=(r_1,r_2,\ldots,r_m)$ and column sums be $\mathcal{C}=(c_1,c_2,\ldots,c_n)$, where

$$n=r_1\geq r_2\geq\ldots\geq r_m,$$

and

$$m=c_1\geq c_2\geq\ldots\geq c_n.$$

Assume that the line sums are consistent. Small-boundary reconstruction conjecture. There exists a set $F_3$ with line sums $(\mathcal{R},\mathcal{C})$ such that

$$L_h(F_3)\leq 4n-4,\qquad L_v(F_3)\leq 4m-4.$$

The conjecture asserts that every consistent pair of line sums admits a reconstruction with simultaneously small horizontal and vertical boundary. The paper proves the corresponding horizontal-boundary bound for its construction, but notes that the analogous vertical bound is not proved and that the construction can have a much larger vertical boundary.